Question: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{-t^2 + 5t + 24}{t^3 - t^2 - 56t}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ a = \dfrac {-1(t^2 - 5t - 24)} {t(t^2 - t - 56)} $ $ a = -\dfrac{1}{t} \cdot \dfrac{t^2 - 5t - 24}{t^2 - t - 56} $ Next factor the numerator and denominator. $ a = - \dfrac{1}{t} \cdot \dfrac{(t - 8)(t + 3)}{(t - 8)(t + 7)}$ Assuming $t \neq 8$ , we can cancel the $t - 8$ $ a = - \dfrac{1}{t} \cdot \dfrac{t + 3}{t + 7}$ Therefore: $ a = \dfrac{ -t - 3 }{ t(t + 7)}$, $t \neq 8$